# Two players alternately name dates

Two players alternately name dates. The winner is the player who names 31st December, and the starting date is 1st January. If a player names Xst of Y Month, the other player can name the first of the next month (1st of Month Y+1) or the next day of the current month (X+1 of Month Y). (For example, the first player begins by naming 1st February or 2nd January). How would one make a strategy for this game?

If the first player is the one who says either 2nd January or 1st February, then the winning player is

the second player

Proof:

The player who says 31st December wins, so the player who says any odd date in December wins, so 1st December also wins.

Therefore, the player who says any November date loses (because the next player can say 1st December).

Therefore, the player saying 31st October wins, so all odd dates in October (also 1st October) win.

So any September date loses.

So 31st August wins, odd dates in August win, and 1st August wins.

So any July loses.

So 30th June wins, even dates in June win, 2nd June wins, 1st June loses.

So 31st May wins, odd dates in May win, and therefore 1st May wins.

Any April date loses.

So 31st March wins, as do all odd dates in March, and therefore 1st March wins.

Any February date loses.

31st January wins, odd dates in January win, so 3rd January wins, and 2nd January loses.

The first player

can only say 2nd January or 1st February, which both lose. Therefore, the second player wins.

The strategy is to

go to the line in which I discussed the current month, and say the next winning date.

Note: I assumed that the game only goes over one year. If players can go on to the next January, then

the game will never end. The reason is that there is no date that forces the next player to say 30th December.